Foundations of Modern Cosmology

Chapter 11: Modeling the Universe

Chapter Summary

A model of the universe includes a mathematical description of how the
scale factor R(t) evolves with time. In this chapter we develop some models
of the universe.

As a first approximation, consider the analogy of the Newtonian ball of
self-gravitating particles. Gravity acts to try to pull the ball together. If
the ball is expanding with sufficient velocity it can resist this collapse.
We obtain a simple equation to describe the evolution of this Newtonian
ball. One of the most important consequences of this analysis is the
realization that gravity permits three possibilities for the evolution of
the universe: it could expand forever; it could stop expanding at infinite
time; or it could stop expanding at some finite point in time and recollapse.

Remarkably, the fully general relativistic solution for a universe consisting
of smoothly distributed matter has the same form as the Newtonian solution.
The equations that describe the evolution of the universe under the influence
of its self-gravity are called the Friedmann equations.
Models in which only gravity operates (i.e. zero cosmological constants) and
mass-energy is conserved are called standard models .
(They are also widely known as Friedmann-Robertson-Walker models, or FRW models,
but we do not refer to them as such in the text.)
The three possible
fates of the universe were seen to correspond to the three basic geometry
types studied in Chapter 8. The hyperbolic universe expands forever; the
flat universe expands but ever more slowly, until it ceases expanding at
infinite time; and the spherical universe reverses its expansion and collapses
in a "big crunch."

From the Friedmann equations we can derive a large number of important
parameters:

the critical density and the Omega parameter

the deceleration parameter q

the relationship of the curvature k to q and Omega

The standard models of cosmology assume zero cosmological constant (Lambda).
The only force acting is gravity. Then we have the following possible models:

Standard Model Summary Table

Model

Geometry

k

Omega

qo

Age

Fate

Closed

Spherical

+1

>1

> 1/2

to < 2/3 tH

Recollapse

Einstein- deSitter

Flat

0

=1

= 1/2

to = 2/3 tH

Expand forever

Open

Hyperbolic

-1

<1

< 1/2

2/3tH < to < tH

Expand forever

The special case of the flat (k = 0, Omega = 1), matter-only universe
is called the Einstein-deSitter model. Various numerical parameters such
as the age of the universe, the lookback time to distant objects, and so
forth, are easiest to compute in the Einstein-deSitter model, so it provides
a convenient guide for estimation of some cosmological quantities.

Adding a nonzero cosmological constant provides a number of new possible
models. Of these nonstandard models, the de Sitter, the Steady State, and
the Lemaitre models are the most significant. The cosmological constant acts
as an additional force, either attractive (negative lambda) or repulsive
(positive lambda). Instead of decreasing in strength with distance like
gravity, the "lambda force" increases with scale factor. This means that
any nonzero cosmological constant will ultimately dominate the universe.
An attractive "lambda force" will cause a recollapse and big crunch regardless
of the model's geometry. However, the possibility of an attractive lambda
in the physical universe is ruled out by observations. A repulsive lambda force
has more interesting possible effects. The details depend upon the model,
but eventually all such models expand exponentially.